Magnetic field driven enhancement of the weak decay width of charged pions

We study the effect of a uniform magnetic field $\vec B$ on the decays $\pi^-\to l^-\bar{\nu}_l$, where $l=e,\,\mu$, carrying out a general analysis that includes four $\pi^-$ decay constants. Taking the values of these constants from an effective model, it is seen that the total decay rate gets strongly increased with respect to the $B=0$ case, with an enhancement factor ranging from $\sim 10$ for $eB=0.1$~GeV$^2$ up to $\sim 10^3$ for $eB=1$~GeV$^2$. The ratio between electronic and muonic decays gets also enhanced, reaching a value of about $1:2$ for $eB=1$~GeV$^2$. In addition, we find that for large $B$ the angular distribution of outgoing antineutrinos shows a significant suppression in the direction of the magnetic field.

The effect of intense magnetic fields on the properties of strongly interacting matter has gained significant interest in recent years [1][2][3].This is mostly motivated by the realization that strong magnetic fields might play an important role in the study of the early Universe [4], in the analysis of high energy non-central heavy ion collisions [5], and in the description of compact stellar objects like the magnetars [6].It is well known that magnetic fields also induce interesting phenomena such as the enhancement of the QCD vacuum (the so-called "magnetic catalysis") [7] and the decrease of critical temperatures for chiral restoration and deconfinement QCD transitions [8].In this work we concentrate on the effect of a magnetic field B on the weak pion-to-lepton decays π − → l − νl .In fact, the study of weak decays of hadrons in the presence of strong electromagnetic fields has a rather long history (see e.g.Refs.[9][10][11][12]).In most of the existing calculations of these decay rates, however, the effect of the external field on the internal structure of the participating particles has not been taken into account.In the case of charged pions, only recently such an effect has been analyzed in the context of chiral perturbation theory [13] and effective chiral models [14][15][16], as well as through lattice QCD (LQCD) calculations [17].An interesting observation has been pointed out in Ref. [17].In that work it is noted that the existence of the background field opens the possibility of a nonzero pion-to-vacuum transition via the vector piece of the hadronic current, implying the existence of a further form factor in addition to the pion decay constant f π (which arises from the axial vector piece).Taking into account this new decay constant and using some approximations for the dynamics of the participating particles, the authors of Ref. [17] obtain an expression for the π − decay width in the presence of the external field.In particular, it is claimed that the decay rate of charged pions into muons could be enhanced by a factor of about 50 with respect to its value at B = 0, for eB ∼ 0.3 GeV 2 .Lately, a more complete analysis of the situation has been presented in Ref. [18], where the most general form of the relevant hadronic matrix elements in the presence of an external uniform magnetic field was determined.It was found that in general the vector and axial vector pion-to-vacuum transitions (for the case of charged pions) can be parametrized through one and three hadronic form factors, respectively.Taking into account all four decay constants, in Ref. [18] an expression for the π − → l − νl decay width that fully takes into account the effect of the magnetic field on both pion and lepton wavefunctions was obtained.The main purpose of this letter is to show that, once these improvements are incorporated, the π − → l − νl decay rate in the presence of the magnetic field turns out to be strongly enhanced with respect to its value for B = 0, the enhancement factor ranging from ∼ 10 for eB = 0.1 GeV 2 up to ∼ 10 3 for eB = 1 GeV 2 .Interestingly, it is found that the ratio between π − partial decay rates into electrons and muons gets also significantly increased, reaching a value of about 0.5 for eB = 1 GeV 2 .In addition, it is observed that already for a magnetic field of eB ≃ 0.1 GeV 2 the angular distribution of the outgoing antineutrinos is expected to be highly anisotropic, showing a significant suppression in the direction of the field.
As well known, in the absence of an external magnetic field the decay width Γ(π − → l − ν) in the pion rest frame is given by where G F is the Fermi effective coupling, θ c is the Cabibbo angle, and the value of the . Owing to the m 2 l factor, the total width is strongly dominated by the muonic decay, for which the branching ratio reaches about 99.99%.The reason for this behavior can be easily understood in terms of helicity suppression.In the pion rest frame, the outgoing charged lepton and antineutrino have opposite momenta, therefore the final state has zero orbital angular momentum, and angular momentum conservation requires both outgoing particles to have opposite spins.Taking the direction of momenta as the angular momentum quantization axis, this implies that the charged lepton l and the antineutrino νl should have the same helicity.On the other hand, the electroweak current couples the π − only to right-handed antineutrinos and left-handed charged leptons.Then, if we assume that neutrinos are massless (which is clearly a good approximation in this case), the helicity of the antineutrino will be +1.In the limit m l → 0 the helicity of the left-handed charged lepton will be −1, hence, the decay turns out to be forbidden if one assumes angular momentum conservation.
In the presence of an external uniform magnetic B, the above situation becomes dramatically modified.For definiteness, let us take the magnetic field to lie along the z axis, B = (0, 0, B), with B > 0. As in the B = 0 case, we assume the charged pion to be in its lowest possible energy state.The latter corresponds to the lowest Landau level (LLL) ℓ = 0, and the pion z component of the momentum p z = 0.It is worth stressing that, even in this lowest energy state, the decaying pion cannot be at rest, due to the existence of a nonvanishing zero-point motion.In fact, the three space components of pion momentum are not a good set of quantum numbers to describe the initial state in this case.Moreover, for nonzero B the canonical angular momentum turns out to be a gauge dependent quantity.
Regarding the pion mechanical angular momentum, it can be shown [20] that the mean field value of its projection on the direction of the magnetic field is 1 for the LLL.Thus, a reasoning similar to that stated for B = 0 does not apply.Since the spins are not fixed to be opposite, the outgoing charged lepton is no longer forced to be right-handed, and the width does not necessarily vanish in the m l → 0 limit.For sufficiently large values of the external field, the only relevant mass scale (besides the Fermi constant) is that given by eB, and the partial decay width becomes approximately independent of the lepton mass.
It is also interesting to analyze the m l → 0 limit in connection with the angular distribution of the outgoing particles.Let us consider an explicit form for the charged lepton state, choosing the Landau gauge A µ = (0, 0, Bx, 0).For the antineutrinos, not affected by the magnetic field, one can simply assume a plane wave state of momentum k.The charged lepton states can be labelled by q = (n, q y , q z ), with energy eigenvalues given by z , where B e = |eB|.The full explicit form of the corresponding wavefunctions can be found e.g. in App.A.3 of Ref. [18].Let us assume that the magnetic field is large enough so that the outgoing charged lepton can only be in the LLL, n = 0 (the validity of this assumption will be discussed below).Considering the explicit form of the corresponding spinor, it is not hard to show that in the limit m l = 0 the eigenstates of the chirality operator satisfy Consequently, states with q z > 0 (q z < 0) are necessarily left-handed (right-handed).Since in the limit m l = 0 there are no lepton chirality-flipping terms in the Lagrangian and the weak current involves only left-handed leptons, only states with q z > 0 are allowed.In this way, the conservation of the z component of total momentum in the decay implies that k z < 0, i.e., the antineutrinos will only come out within the half-space z < 0. Thus, contrary to the situation at B = 0, in the limit m l → 0 and large eB one does not expect any "helicity suppression" but a rather significant anisotropy in the angular distribution of the outgoing antineutrinos.
To quantitatively see how important this "non-helicity suppression" effect is, one has to analyze in detail the π − → lν decay width in the presence of the magnetic field.A model-independent expression for the width has obtained in Ref. [18], considering a pion in the LLL state (ℓ = 0) with p z = 0.The result, expressed in terms of three form factor combinations, is given by where The function A where We note here that in Eq. (3) the integration variable u is proportional to the antineutrino transverse momentum squared, k 2 ⊥ = k 2 x + k 2 y , while an integration over the parallel component k z has already been performed.
The decay constants in Eq. ( 6), defined in Ref. [18], parametrize the most general form of the pion-to-vacuum vector and axial vector hadronic matrix elements.Their theoretical determination would require either to use LQCD simulations or to rely on some hadronic effective model.Before addressing possible estimates for these quantities, let us analyze how "non-helicity suppression" is realized in Eq. ( 3).Once again we concentrate in the case of a large external magnetic field.Since the pion is built of charged quarks, the pion mass will depend in general on the magnetic field.Now, if the mass growth is relatively mild, for large magnetic fields one should get B e > m 2 π − m 2 l .In fact, this is what one obtains from lattice QCD calculations [17] as well as from effective approaches like the Nambu-Jona-Lasinio (NJL) model [16], for values of B e say 0.05 GeV 2 .Then, according to the above expressions, this implies n max = 0, and the outgoing muon or electron (let us assume that the energy is below the τ production threshold) is expected to lie in its LLL (n = 0).A further simplification can be obtained when the squared lepton mass can be neglected in comparison with B e (or, equivalently, in comparison with E 2 π − , which is expected to grow approximately as B e ).For m l ≪ B e , one can take m l → 0.Then, E l = kz and the integral over k ⊥ extends up to E π − .In this limit the decay width is given by As anticipated, there is no helicity suppression, and the width does not vanish in the m l = 0 limit.In fact, it turns out to grow with the magnetic field as B 2 e /E π − , with some suppression due to the factor in square brackets.Clearly, the relevance of Eq. ( 7) depends on whether the form factor combination on the right hand side is nonnegligible for eB much larger than m 2 l .While this is likely to happen for the π − decay to e − νe , in the case of the muon (and, of course, the tau) the situation is less clear, and corrections arising from a nonzero lepton mass should be taken into account.
In order to provide actual estimates for the magnetic field dependence of the π − decay width we need some input values for the decay constants.Although some results have been provided by existing LQCD simulations [17], present lattice analyses do not include all the constants appearing in Eq. ( 6).Therefore we will consider here the values calculated in Ref. [21] within the framework of the NJL model.Our results, shown in Fig. 1, correspond to the parameter set denoted by "Set I" in Ref. [21].In the left panel we quote the π − partial decay widths to both µ − νµ and e − νe as functions of eB, in a logarithmic scale.It is seen that the partial widths become strongly enhanced when the magnetic field is increased above say 0.1 GeV 2 /e.This enhancement is more pronounced for the decay to e − νe (dashed line), since for low values of B helicity suppression becomes important.The bump observed in this curve for eB ∼ 10 −2 GeV 2 is due to the fact that this region is dominated by the n = 1 Landau level contribution, which disappears at about eB ∼ 2 × 10 −2 GeV 2 leaving n = 0 as the only energetically allowed electron Landau level.The dotted line in the graph corresponds to the asymptotic decay width quoted in Eq. ( 7).In the central panel we show the behavior of the total decay width Γ e + Γ µ , normalized to its value at B = 0.For this effective model the enhancement factor is found to be about 1000 for eB ≃ 1 GeV 2 .Finally, in the right panel we quote the ratio Γ e /Γ µ as a function of eB (notice that in this plot scales are linear).The absence of helicity suppression leads to a strong increase of this ratio with the magnetic field, reaching a value of about 0.5 for eB ≃ 1 GeV 2 , while for B = 0 one has Γ e /Γ µ ≃ 1.2 × 10 −4 .It is worth mentioning that the results in Fig. 1 do not depend significantly on the model parametrization (e.g. it is seen that the results for parameter Sets II and III of Ref. [21] do not differ from those in Fig. 1 by more than 3%).It is also interesting to discuss with some detail the angular distribution of the outgoing antineutrinos.As stated, while for B = 0 the distribution is isotropic, this should change significantly in the presence of a large magnetic field.Denoting w = cos θ = k z /| k|, the differential decay rate can be written as where and the function B The term proportional to B (n) π − (u) in Eq. ( 8) vanishes after integration over w, therefore it does not contribute to the total decay width.
Once again, to get definite predictions for the angular distributions we rely on the values for the pion mass and decay constants obtained in Ref. [21] within the NJL model, taking the parameter Set I. Our numerical results for the normalized differential partial decay widths are shown in Fig. 2, where several representative values of eB are considered.Left and right panels correspond to π − decays into e − νe and µ − νµ , respectively.It is seen that the fraction of antineutrinos that come out in the half-space w > 0 fluctuates when the magnetic field is increased, becoming strongly suppressed for values of eB much larger than the lepton mass squared.This is consistent with the discussion below Eq. ( 2), in which it is stated that for large B in the m l = 0 limit no antineutrinos should be produced in the direction of the magnetic field.Indeed, for m l = 0, assuming that B is large enough so that the lepton LLL is the only one allowed in the final state, the normalized differential decay width is given by where λ = E 2 π − /(2B e ).In addition, it is worth noticing that for large values of B most antineutrinos come out with low |k z |, i.e. in directions approximately perpendicular to the magnetic field.
In summary, in this letter we get an estimation of the effect of an external uniform magnetic field on the magnitude of the decay rate Γ(π − → l − νl ) and the angular distribution of the antineutrinos in the final state.Our analysis takes into account the contribution of all four possible π − decay form factors.The values of these constants and that of the pion mass are taken from a NJL model for effective strong interactions, considering the π − in its lowest possible energy state.Our results show that the total decay rate Γ e + Γ µ becomes strongly increased with respect to its value at B = 0, the enhancement factor ranging from ∼ 10 for eB = 0.1 GeV 2 up to ∼ 10 3 for eB = 1 GeV 2 .Moreover, owing to the presence of the new decay constants and the features of nonzero B kinematics, it is found that the decay width Γ l does not vanish in the limit m l = 0.As a consequence, for large values of B the ratio Γ e /Γ µ changes dramatically with respect to the B = 0 value (of about 1.2 × 10 −4 ), reaching a magnitude of ∼ 0.5 at eB ≃ 1 GeV 2 .This could be interesting e.g.regarding the expected flavor composition of neutrino fluxes coming from the cores of magnetars and other stellar objects.Finally, it is found that for large B the angular distribution of outgoing antineutrinos is expected to be highly anisotropic, showing a significant suppression in the direction of the external field.

2 ]Figure 1 :
Figure 1: (Color online) Left panel: π − partial decay widths into e − νe (dashed line) and µ − νµ (full line), and n = 0 asymptotic contribution for m l = 0 (dotted line) as functions of eB.Central panel: total decay width as a function of eB, normalized to its value at B = 0. Right panel: ratio Γ e /Γ µ as a function of eB.The results correspond to the model in Ref. [21], parameter Set I.

2 Figure 2 :
Figure 2: (Color online) Normalized differential partial decay widths of the π − into e − νe (left) and µ − νµ (right), as functions of w = cos θ for selected values of eB.The results correspond to the model in Ref [21], parameter Set I.